A parametric array in air results from the introduction of sufficiently intense, audio modulated ultrasonic signals into an air column. Self demodulation, or down-conversion, occurs along the air column resulting in an audible acoustic signal. This process occurs because of the known physical principle that when two sound waves with different frequencies are radiated simultaneously in the same medium, a sound wave having a wave form including the sum and difference of the two frequencies is produced by the non-linear interaction (parametric interaction) of the two sound waves. So, if the two original sound waves are ultrasonic waves and the difference between them is selected to be an audio frequency, an audible sound is generated by the parametric interaction. However, due to the non-linearities in the air column down-conversion process, distortion is introduced in the acoustic output. The distortion can be quite severe and 30% or greater distortion may be present for a moderate modulation level. Lowering the modulation level lowers the distortion, but at the expense of both a lower output volume and a lower power efficiency.
In 1965, Berktay formulated that the secondary resultant output (audible sound) from a parametric loudspeaker is proportional to the second time derivative of the square of the modulation envelope. It was shown by Berktay that the demodulated signal, p(t), in the far-field is proportional to the second time derivative of the modulation envelope squared.p(t)∝∂2/∂t2[(env(t))2].  (Equation 1)This is called “Berktay's far-field solution” for a parametric acoustic array. Berktay looked at the far-field because the ultrasonic signals are no longer present there (by definition). The near-field demodulation produces the same audio signals, but there is also ultrasound present which must be included in a general solution. Since the near-field ultrasound isn't audible, it can be ignored and with this assumption, Berktay's solution is valid in the near-field too.
The earliest use of this relationship for parametric loudspeakers in air was a modulator design for parametric loudspeakers in 1985. This advancement included the application of a square root function to the modulation envelope. Using the square root function compensates for the natural squaring function which distorts the envelope of the modulated sideband signal emitted to the air. Those skilled in the art have also shown that the square root double sideband signal can theoretically produce a low distortion system but at the cost of infinite system and transducer band width. It is not practical to produce any device that has an infinite bandwidth capability. Further, the implementation of any significant bandwidth means that the inaudible ultrasonic primary frequencies will, on the lower sideband, extend down into the audible range and cause new distortion which is at least as bad as the distortion eliminated by the infinite bandwidth square root pre-processing system.
In a typical application, the desired signal is amplitude modulated (AM) modulated on an ultrasonic carrier of 30 kHz to 50 kHz, then amplified, and applied to an ultrasonic transducer. If the ultrasonic intensity is of sufficient amplitude, the air column will perform a demodulation or down-conversion over some length (the length depends, in part, on the carrier frequency and column shape). The prior art, such as U.S. Pat. No. 4,823,908 to Tanaka, et al., teaches that the modulation scheme to achieve parametric audio output from an ultrasonic emission uses a double sideband signal with a carrier frequency and sideband frequencies spaced on either side of it by the frequency difference corresponding to the audio frequencies of interest.
For example, when amplitude modulating a 6 kHz tone onto a 40 kHz carrier, as shown in FIG. 1, sideband frequencies are generated. FIG. 2 shows that the carrier frequency (40 kHz) is now accompanied by a 34 kHz lower-sideband and a 46 kHz upper-sideband. Three components are now present, 34 kHz, 40 kHz, and 46 kHz which gives a pure 6 kHz envelope. As described previously, the 6 kHz signal would be square rooted before being used as the modulation signal shown in FIG. 3. Using a spectrum produced by the square root function for the modulation signal of a 40 kHz carrier generates the spectral components shown in FIG. 4. Applying a square root function to the 6 kHz signal produces infinite harmonics, and the AM spectrum has upper and lower sideband frequencies that are also infinitely far from the carrier. It is infeasible to implement this type of system because of transducer bandwidth limitations and similar problems.
In practice, the first five or six harmonics are enough to give a good approximation of the ideal square rooted wave. However, even when the number of harmonics is limited, the low sideband frequencies still reach down into the audio range and create distortion. As in the foregoing example in FIGS. 1-4, the lower-sideband frequencies that would need to be emitted are 34, 28, 22, 16, 10 and 4 kHz. This creates the problem that audible frequencies (16, 10, and 4 kHz) will need to be emitted along with the ultrasonic ones to make the desired modulation envelope.
Applying a square root function to the original signal reduces or eliminates the distortion in the demodulated audio but it creates unwanted audible frequencies that are emitted. In the current state of the prior art, the only choice is between high distortion (avoiding the square root function) or a wide bandwidth requirement with less distortion (using a square root function). Further, the square rooted signal for any given ultrasonic frequency is only valid for low level signals. As the ultrasonic power levels are increased to provide significant audio output, the ideal envelope shifts from the square root of the signal to the audio signal itself (or 1 times the signal).
Another problem exhibited by parametric loudspeaker systems is that as the frequency and/or intensity of the ultrasonic sound waves is increased to allow room for lower sidebands and to achieve reasonable conversion levels in the audible range, the air can be driven into saturation. This means that the fundamental ultrasonic frequency is limited as energy is robbed from it to supply the harmonics. The level at which the saturation problem appears is reduced 6 dB for every octave the primary frequency is increased. In other words, the power threshold at which saturation appears, decreases as the frequency increases. Double sideband signal systems used with parametric arrays must always be at least the bandwidth of the signal above any audible frequency (assuming a 20 kHz bandwidth) and even more if the distortion reducing square root function is used which also demands an infinite bandwidth.
A further problem with prior art parametric loudspeakers is that they have a built in high pass filter characteristic such that the amplitude of the secondary signal (audio output) falls at 12 dB per octave for descending frequencies. Because the lower sideband of a double sideband system must be kept from producing output in the audible range, the carrier frequency must be kept at least 20 kHz above the audible upper limit for double sideband (DSB) and at the very least twice that amount with a square rooted DSB. This range forces the carrier frequency up quite high. As a result, the saturation limit is easily reached and the overall efficiency of the system suffers.
These excessive and undesirable types of distortion preclude the practical or commercial use of the uncompensated parametric arrays or even square-rooted compensation schemes in high fidelity applications. Accordingly, it would be an improvement over the state of the art to provide a new method and system for pre-processing the audio signal which would result in lowered distortion with a decreased bandwidth requirement for the ultrasonic parametric array output. It would also be desirable to use lower primary frequencies which are still above the audible range to produce less saturation and attenuation.